a one-parameter subgroup

Pick a steady, unchanging 'rate of symmetry' and let it run: a constant angular velocity gives a clock-hand sweeping around, a constant boost gives uniform acceleration, a constant translation rate gives steady drift. Each of these is a one-parameter family of group elements indexed by time, where time adds: running for s seconds then t more is the same as running for s + t. That is a one-parameter subgroup — the trajectory of a single, unchanging infinitesimal symmetry.

Formally, a one-parameter subgroup of a Lie group G is a smooth homomorphism gamma: R -> G from the additive group of real numbers, so gamma(s + t) = gamma(s) * gamma(t) and gamma(0) = e. Differentiating at 0 gives a vector X = gamma'(0) in the Lie algebra g, the 'initial velocity'. The fundamental theorem is that this correspondence is a bijection: every X in g determines a unique one-parameter subgroup, namely the integral curve through e of the left-invariant vector field with value X at the identity, and conversely each one-parameter subgroup is recovered this way. Concretely, gamma satisfies the ODE gamma'(t) = (L_gamma(t))_* X with gamma(0) = e; the solution is gamma(t) = exp(tX). On a matrix group this is the matrix exponential, gamma(t) = e^(tA) = I + tA + (tA)^2/2! + ..., and indeed e^(sA) e^(tA) = e^((s+t)A).

One-parameter subgroups are the bridge from the Lie algebra to the Lie group: they package each infinitesimal generator X into an honest curve of group elements, and assembling all of them gives the exponential map. They are exactly the flows of left-invariant vector fields, which is why those fields are complete. A caution: the image of gamma need not be a closed subset of G — the famous irrational line on the torus T^2 = R^2/Z^2 is a one-parameter subgroup whose image winds densely and is not closed, so a one-parameter 'subgroup' is an immersed, not necessarily embedded, submanifold.